1. Lecture 12 Security Market Line (SML)

2. Topics covered 1.Definitions 2.Beta – definition and use 3.SML formula 4.Steps to drawing the SML 5.Interpreting the SML 6.SML and the reward-to-risk ratio 7.Two cases of disequilibrium 8.Numerical examples on using the SML 9.Practice Question

3. Definitions Market portfolio = a portfolio that consists of most, if not all, of the assets in the market Risk-free asset = an asset such as a government-issued T-Bill that is, to all intents and purposes, risk- free in the sense that they are not subject to the same probability of default as other assets in our investment world Market return = rate of return on market portfolio = R M Risk-free rate = rate of return on the risk-free asset = R f Risk premium on any asset i = Return on asset i – Risk-free rate = R i – R f Market risk premium = Market return – Risk-free rate = R M – R f

4. Beta Beta (  ) – a measure of systematic risk of an asset i relative to an average asset representing the market portfolio  M = 1 - beta or systematic risk on the average asset representing the market portfolio (hereafter referred to as beta of market portfolio), set as 1  f = 0 - beta or systematic risk on risk-free asset, set as 0

5. What use  ?  is useful when we want to compare the risk of different assets. Higher  => Higher systematic risk Example:  X = 0.5 => Asset X is half as risky as the market portfolio  Y = 2 => Asset Y is two times as risky as the market portfolio

6. SML Formula and interpretation SML Formula: R i = R f +  i (R M – R f ) Interpretation: The return on any asset i (R i ) depends on -the return on the risk-free asset, R f -the systematic risk of asset i,  i -the market risk premium, R M – R f

7. Graphing the SML – Step 1 Draw the x-axis and y-axis:

8. Graphing the SML – Step 2 Label the x-axis as  i and the y-axis as R i :

9. Graphing the SML – Step 3 Draw in a diagonal straight-line: ii RiRi 0

10. Graphing the SML – Step 4 Label the point where the line crosses the y-axis (generally called the intercept) as R f : ii RiRi 0

11. Graphing the SML – Step 5 Label the line as R i = R f +  i (R M – R f ): ii RiRi 0 RfRf

12. Interpreting the SML: Return and beta on market portfolio (R M and  M ) ii RiRi 0 RfRf R i = R f +  i (R M – R f )

13. Interpreting the SML: Slope ii RiRi 0 RfRf R i = R f +  i (R M – R f ) RMRM MM

14. Reward-to-risk ratio Reward-to-risk ratio of asset i = (R i – R f ) /  i SML => the reward-to-risk ratio for all assets must be the same at equilibrium

15. Disequilibrium Case #1: If asset return plot above the SML ii RiRi 0 RfRf R i = R f +  i (R M – R f )

16. Disequilibrium Case #2: If asset return plot below the SML ii RiRi 0 RfRf R i = R f +  i (R M – R f )

17. Using the SML Given R f,  i, and R M – R f => can find R i Given R f, R i, and R M – R f => can find  i

18. Numerical Example: Finding R i Given that the risk-free rate is 5% and the market risk premium is 6%, what is the equilibrium return on an asset with a beta of 2?

19. Market risk premium vs. Market portfolio return Market risk premium Market portfolio return Given that the risk-free rate is 5% and the market return is 11%, what is the equilibrium return on an asset with a beta of 2?

20. Numerical Example: Finding  i Given that the risk-free rate is 3% and the market risk premium is 7%. a.What is the beta for Asset A with a return of 12%? b.What is the beta for Asset B with a return of 8%? c.Which asset, A or B, has the higher systematic risk? d.What are the reward-to-risk ratios for Assets A and B?

21. Numerical Example: Finding  i

22. There can never be too many Practice Questions There are two assets in the market. Asset X has a return of 15% and a beta of 1.9. Asset Y has a return of 8% and a beta of 0.9. Use the SML formula to find the risk-free rate (R f ), the market risk premium (R M - R f ), and the market portfolio return (R M ). Check answer on next 2 pages.

23. Answer to Practice Question Information given: Rx = 0.15  x = 1.9 Ry = 0.08  y = 0.9 SML for X: R x = R f +  x (R M – R f ) Plugging in the numbers for X, we get 0.15 = R f + 1.9(R M – R f ) SML for Y: R y = R f +  y (R M – R f ) Plugging in the numbers for Y, we get 0.08 = R f + 0.9(R M – R f ) We have two equations and two unknowns (R f and (R M – R f )). We can solve for them mathematically.

24. Answer to Practice Question (cont.) Solving 2 equations and 2 unknowns: Eq1: 0.15 = R f + 1.9(R M – R f ) Eq2: 0.08 = R f + 0.9(R M – R f ) Subtract Eq2 from Eq1: 0.15 – 0.08 = R f - R f + 1.9(R M – R f ) - 0.9(R M – R f )  0.07 = (1.9 – 0.9)(R M – R f )  0.07 = 1(R M – R f ) Dividing both sides by 1, we get (R M – R f ) = 0.07/1 = 0.07 Plugging (R M – R f ) = 0.07 into Eq1, we get 0.15 = R f + 1.9(0.07)  0.15 = R f + 0.133 Subtracting both sides by 0.133, we get R f = 0.15 – 0.133 = 0.017 With (R M – R f ) = 0.07 and R f = 0.017, we can easily find the market portfolio return: RM = (R M – R f ) + R f = 0.07 + 0.017 = 0.087 Check: SML for X: Rx = 0.017 + 1.9(0.07) = 0.15 SML for Y: Ry = 0.017 + 0.9(0.07) = 0.08 Summary of Results: R f = 1.7%, R M = 8.7%, R M – R f = 7%

25. End of Lecture 12 on SML